Enigma 1635: Marty’s date

Our niece Ennie, nephew Iggy and their friend Marty were born on different dates in December. Ennie’s birth date is the 19th and Iggy’s is the 21st. The trio have tried writing down all numbers that can be expressed as a multiple of 19 plus a multiple of 21 plus a multiple of Marty’s birth date, where in each case the whole-number multiple can be zero or more.

They have managed to express 189 as a sum of multiples, and indeed any higher number can be similarly expressed. But it is impossible to express 188 in this way.